D O I: 10.1051/0004-6361/201526330
© E S O 2015
Astronomy &
Astrophysics
The VIMOS Public Extragalactic Redshift Survey
Reconstruction of the redshift-space galaxy density field*
B. R. Granett1’**, E. Branchini2,3,4, L. Guzzo1,5, U. Abbas6, C. Adami7, S. Arnouts8, J. Bel1, M. Bolzonella9, D. Bottini10, A. Cappi9,11, J. Coupon12, O. Cucciati13,9, I. Davidzon9,13, G. De Lucia14, S. de la Torre7, A. Fritz10,
P. Franzetti10, M. Fumana10, B. Garilli10,7, O. Ilbert7, A. Iovino1, J. Krywult15, V. Le Brun7, O. Le Fevre7, D. Maccagni10, K. Małek16, F. Marulli13,17,9, H. J. McCracken18, M. Polletta10, A. Pollo19,16, M. Scodeggio10, L. A. M. Tasca7, R. Tojeiro20, D. Vergani21,9, A. Zanichelli22, A. Burden23, C. Di Porto9, A. Marchetti10, C. Marinoni24,
Y. Mellier18, T. Moutard4, L. Moscardini13,17,9, R. C. Nichol23, J. A. Peacock25, W. J. Percival23, and G. Zamorani9
(Affiliations can be found after the references) Received 16 April 2015 / Accepted 19 June 2015
A B STR A CT
Aims. Using the VIMOS Public Extragalactic Redshift Survey (VIPERS) we aim to jointly estimate the key parameters that describe the galaxy density field and its spatial correlations in redshift space.
Methods. We use the Bayesian formalism to jointly reconstruct the redshift-space galaxy density field, power spectrum, galaxy bias and galaxy luminosity function given the observations and survey selection function. The high-dimensional posterior distribution is explored using the Wiener filter within a Gibbs sampler. We validate the analysis using simulated catalogues and apply it to VIPERS data taking into consideration the inhomogeneous selection function.
Results. We present joint constraints on the anisotropic power spectrum, and the bias and number density of red and blue galaxy classes in luminosity and redshift bins as well as the measurement covariances of these quantities. We find that the inferred galaxy bias and number density parameters are strongly correlated although they are only weakly correlated with the galaxy power spectrum. The power spectrum and redshift- space distortion parameters are in agreement with previous VIPERS results with the value of the growth rate f a 8 = 0.38 with 18% uncertainty at redshift 0.7.
Key words. large-scale structure of Universe - cosmology: observations - galaxies: statistics - cosmological parameters - methods: statistical - methods: data analysis
1. Introduction
T he distribution o f galaxies on large scales provides a fu n d a
m ental test o f the cosm ological m odel. In th e standard picture, galaxies trace an underlying m atter density field an d the sta
tistical properties o f this field such as its pow er spectrum and higher o rder m om ents are given b y the theory (P eebles 1980) . This clear view is confounded, however, by the sparse distribu
tions o f lum inous galaxies m ap p ed by surveys (L ahav & Suto 2004) . G alaxies are com plex system s; they are biased tracers o f the n on-linear m atter field and their clustering strength depends on their properties and form ation histories (D avis & G eller 1976;
B lanton et al. 2 0 0 5 ; K aiser 1984; B ardeen e t al. 1986; M o &
W hite 1996) . F urtherm ore, their redshift gives a distorted view o f distance, w hich is affected by coherent and ran d o m velocities (K aiser 1987) .
U pcom ing galaxy surveys w ill be sensitive to subtleties in these trends requiring increasingly sophisticated m odelling and num erical sim ulations to interpret th e galaxy distribution in d e
tail (T he D ark E nergy S urvey C ollaboration 2 0 0 5 ; L S S T D ark E nergy S cience C ollaboration 2 0 1 2 ; L evi et al. 2 0 1 3 ; Laureijs et al. 2011) . A t the sam e tim e, these surveys w ill b e sufficiently
* Appendices are available in electronic form at h ttp ://w w w .a a n d a .o rg
** Corresponding author: Benjamin R. Granett, e-mail: b e n . g r a n e t t @ b r e r a .i n a f . i t
large to b e lim ited b y m inute selection effects that system ati
cally and significantly alter th e observed distribution o f galaxies.
Instrum ental and observational artefacts can m asquerade as g en u in e astrophysical effects and vice-versa. Thus the analyses w ill n ee d to track a large nu m b er o f instrum ental and astrophysical param eters and b e able to characterise th e covariances betw een them . R eliable error estim ation w ill req u ire incorporating the set o f both system atic an d ran d o m uncertainties. T he stakes are high as experim ents pro m ise highly p recise constraints on th e n ature o f gravity, dark energy and dark m a tte r (A m endola et al. 2013) .
Together, the physical and instrum ental m odels com pose the total d ata m odel. G iven the large n um ber o f param eters, the B ayesian approach is often p referred over the frequentist one to set jo in t constraints on th e relevant physical quantities (T rotta 2008) . A t the heart o f this approach is the B ayes theorem w hich dictates a recipe for translating a set o f observations into co n straints on m odel param eters. O f fundam ental im portance is the incorporation o f any prio r know ledge o f these param eters. This fram ew ork provides a natural m eans to jo in tly constrain physical param eters o f interest w hile m arginalising over a set o f nuisance param eters. A p aradigm atic exam ple is the analysis o f cosm ic m icrow ave b ackground data (Jew ell et al. 2 0 0 4 ; E riksen et al.
2 0 0 4 ; W andelt et al. 2004) as dem onstrated through the E SA P lanck m ission results (P lanck C ollaboration I. 2015) .
T he W iener filter is th e first exam ple o f the application o f B ayesian reconstruction techniques to galaxy surveys. The
Article published by EDP Sciences A61, page 1 of 16
W iener solution corresponds to the m axim um a posteriori so
lution given a G aussian likelihood an d prior. In general, for a signal contam inated by noise, the W iener filter gives a rec o n struction o f the true signal w ith the m inim um residual variance (R ybicki & Press 1992) . This is also true for non-G aussian sig n al and n oise sources, and fo r this reason, since the galaxy field is n o t G aussian (it is thought to tend tow ard G aussianity on very large scales), the W iener filter has seen significant use in rec o n structing the density field from galaxy surveys (e.g. L ahav e t al.
1994) and in p articular to p redict large-scale structures behind the G alactic plane (Z aroubi et al. 1995) .
W iener filtering is com parable to other adaptive density re construction techniques such as D elaunay tessellations (S chaap
& van de W eygaert 2000) , although W iener filtering offers the advantage o f n aturally accounting for a com plex survey selec
tion function w ith inhom ogeneous sam pling. E xam ples o f ap p li
cations o f W iener filtering to galaxy surveys include th e Two- degree F ield G alaxy R edshift S urvey (2dFG R S) in w hich the W iener filter w as u sed to identify galaxy clusters and voids (E rdogdu et al. 2004) . K itaura e t al. (2009) p resent a W iener d en sity field reconstruction o f the S loan D igital Sky Survey (SD SS) m ain sam ple. A pplied to the V IM O S E x tragalatic R edshift Survey (V IPER S; G uzzo et al. 2014), the W iener filter can n a t
urally account for inhom ogeneous sam pling and survey gaps.
In a com parison study o f different density field estim ators for V IPER S C ucciati et al. (2014) find th at the W iener filter p er
form s w ell although it over-sm ooths the field in low -density environm ents affecting cell-count statistics.
P hysically m otivated p robability distribution functions have been developed to im prove on the W iener filter and obtain u n b i
ased density field reconstructions. K itaura e t al. (2010) dem o n strate in a com parison study th at the use o f a Poisson sam pling m odel fo r th e galaxy counts w ith a log-norm al p rio r on the d en sity field allow s b etter estim ation o f th e low est and hig h est d en sity extrem es on sm all scales. T he generalisation o f th e m odel calls for a fully non-linear solver (Jasche & K itaura 2010) . The Poisson log-norm al m odel was used to reco n stru ct th e density field p ro b ed b y the SDSS sam ple (Jasche et al. 2010a) .
T he G aussian likelihood has also been used to construct m axim um a posteriori estim ators for th e galaxy pow er spectrum (E fstathiou & M oody 2 0 0 1 ; Tegm ark e t al. 2 0 0 2 ; P ope et al.
2 0 0 4 ; G ranett e t al. 2012) . F or the g alaxy lum inosity function e s
tim ates, m axim um likelihood techniques have also enjoyed sig nificant use (Ilbert e t al. 2 0 0 5 ; B lanton e t al. 2 0 0 3 ; E fstathiou et al. 1988) .
G aussian likelihood m ethods have only recently been devel
o ped to jo in tly in fer the density field, po w er spectrum and lu m inosity function from galaxy surveys (K itaura & EnBlin 2 0 0 8 ; EnBlin et al. 2009) . T he first application to the S loan D igital Sky Survey was dem onstrated b y Jasche et al. (2010b) w ho utilise a G aussian likelihood and p rio r to jo in tly estim ate the und erly ing galaxy field an d pow er spectrum . This w ork w as generalised to sim ultaneously estim ate the lin e ar galaxy bias an d lum in o s
ity function (Jasche & W andelt 2013b) . A ta et al. (2015) fu r
th e r m odel a scale-dependent an d stochastic galaxy bias using the log-norm al Poisson m odel. T he m ethodology has also been developed fo r p hotom etric red sh ift surveys (Jasche & W andelt 2012) as first p ro p o sed by K itaura & EnBlin (2008) .
T he p ec u lia r velocities o f galaxies disto rt the density field in ferred from red sh ift surveys (K aiser 1987) . T he average ef
fect m ay b e accounted for b y a convolution operation (L andy &
Szalay 2002) . This serves on large scales w here the density field and velocity field m ay be inferred in a self-consistent m an n er (K itaura et al. 20 1 2 b ; N u sser & D avis 1994) .
T he full description o f th e galaxy field requires co n sidera
tion o f th e h igher o rder m om ents and depends on the physics o f structure form ation. Thus reconstruction m ethods have b een d e
v eloped th at incorporate p hysical m odels b ased on second-order p erturbation theory (K itaura et al. 2 0 1 2 a; K itaura 2 0 1 3 ; Jasche
& W andelt 2 0 1 3 a; Jasche e t al. 2015) o r approxim ate n-body m ethods such as the particle-m esh code (W ang e t al. 2014) . R econstructions o f th e local U niverse have been used in novel w ays, including to estim ate th e bias in the H ubble co nstant due to cosm ic flows (H ess & K itaura 2014) .
In this w ork w e carry out a B ayesian analysis o f the V IM O S E x tragalactic R edshift S urvey (V IPER S; G uzzo et al. 2014). O ur goal is to jo in tly estim ate th e key statistics including the m atter po w er spectrum , galaxy biasing function an d galaxy lum inosity function. O u r strategy is, given the observed n u m b er density o f galaxies in the survey as a function o f position N (R A , D ec, z), to com pute the conditional p robability distribution fo r the param eters, w ritten schem atically as: the m atter over-density field 6, galaxy m ean nu m b er density N galaxy bias b and the tw o-point correlation function S . T he conditional probability distribution or p o sterio r m ay b e decom posed using B ayes theorem :
p(6, N , b, S |N) k p(N |6, N , b, S )p(6, N , b, S ). (1) T he first and second factors on the right h and side are th e data likelihood and the p aram eter prior. W e w ill account for obser
vational system atics such as the survey selection function in th e data m odel, but w e w ill n o t propagate th eir uncertainties.
F o r V IPE R S, the uncertainties in the selection function are sub
dom inant com pared w ith th e statistical errors and so th e in clu sion o f the uncertainties w ill b e reserved fo r future w ork. W e ad o p t th e G ibbs sam pling algorithm to sam ple from th e p o ste
rio r distribution (M arin & R obert 2007) . W ith this approach the com plex jo in t p robability distribution is broken up in a nu m b er o f sim pler, individual conditional distributions. Sam pling these distributions allow s us to build up a M arkov chain th at rapidly converges to the jo in t distribution.
V IPER S has m ap p ed the galaxy field to red sh ift 1 w ith un precedented fidelity (G uzzo et al. 2 0 1 4 ; G arilli et al. 2014) . So far, V IPER S data have been used to constrain the grow th rate o f structure through the shape o f th e redshift-space galaxy correla
tion function (de la Torre et al. 2013) . T he cosm ological inter
p retation o f the galaxy po w er spectrum m o nopole has been p re sented by R o ta et al. (in prep.). T he biasing function th at links th e galaxy and d ark m a tte r density has been estim ated using the lu m inosity-dependent correlation function (M arulli et al. 2013) an d th e shape o f the one-point probability distribution func
tion o f galaxy counts in cells (D i Porto e t al. 2014) . M oreover V IPER S has tightened th e constraints on th e galaxy lum inos
ity and stellar m ass functions (D avidzon et al. 2 0 1 3 ; F ritz et al.
2014) . T hese m easurem ents firm ly anchor m odels o f galaxy form ation at red sh ift 1.
B eyond th e one- an d tw o- poin t statistics o f th e galaxy field, galaxies are organised into a cosm ic w eb o f knots, filam ents and w alls th at surround large em pty voids. In V IPER S the higher or
d e r m om ents o f the galaxy counts in cells distribution function have been m easured (C appi e t al. 2015) . W hile ongoing efforts are being m ade to m easure the m orphologies o f cosm ic struc
tures. A catalogue o f voids has been constructed w ith V IPER S (M icheletti et al. 2 0 1 4 ; H aw ken e t al., in prep.). M o re gener
ally, the M inkow ski functionals can be used to characterise the topology o f large-scale structure as a function o f scale (Schim d, in prep.). T hese m easurem ents typically call for p recise re constructions o f th e density field co rrected for observational system atics such as survey gaps and inhom ogeneous sam pling.
This w ork extends previous analyses b y considering the jo in t distribution o f galaxy lum inosity, colour and clustering bias w ith the spatial p ow er spectrum an d density field. W e b egin in Sect. 2 w ith an overview o f V IP E R S and the p aram eterisation o f the selection function. T he data m o d el is described in Sect. 3 , and the m eth o d is outlined in Sect. 4 . In Sect. 5 w e present the co n straints from th e V IP E R S data.
W e assum e the follow ing fiducial cosm ology Q m = 0.27, Q b = 0.0469, O a = 0.73, n s = 0.95, H 0 = 70 k m s - 1 M pc,
^ 8 = 0.80. This coincides w ith the M ultiD ark sim ulation run (P rada et al. 20 1 2 ) that w as used to construct the m o c k V IPER S catalogues. M agnitudes are in the A B system unless noted. The absolute m agnitudes used w ere com puted under a flat cosm ology w ith Q m = 0.30; how ever w e transform all m agnitudes to the Q m = 0.27 cosm ology.
F ig .1. VIPERS galaxy number density of the v5 internal release sam
ple. The curves show the effect of the completeness corrections includ
ing the spectroscopic success rate (SSR) and target sampling rate (TSR).
Our analysis uses the redshift range 0.6-1.0 (vertical lines).
2. VIPERS
T he V IM O S P ublic E xtragalactic R edshift S urvey (V IP ER S) is an E S O p rogram m e on VLT (E uropean S outhern O bservatory - Very L arge Telescope; G uzzo e t al. 2 0 1 4 ; G arilli et al. 2014) . T he survey targets galaxies for m edium resolution spectroscopy using V IM O S (V Isible M u lti-O bject S pectrograph; L e Fevre et al. 2003) w ithin tw o regions o f the W 1 an d W 4 fields o f the C F H T L S -W ide Survey (C anada-F rance-H aw aii Telescope L egacy W id e;C uillandre et al. 2012) . Targets are chosen b ased upon colour selection to be in the red sh ift range 0 .5 -1 .2 . The final expected sky coverage o f V IPER S is 24 deg2.
F or each galaxy, the B -band rest-fram e m agnitude w as esti
m ated follow ing the Spectral E nergy D istribution (SED ) fitting m eth o d d escribed in D avidzon e t al. (2013) and adopted to d e
fine volum e lim ited sam ples. T he choice o f B -band rest-fram e is natural, corresponding to th e observed I-b a n d at red sh ift ~0.8.
W e derived K -corrections from the best-fitting SED tem plates using all available photom etry including near-UV, optical, and near-infrared.
2 .1 . S a m p l e s e l e c ti o n
This analysis is b ased on the V IPER S v5 internal d ata release w hich represents 77% o f the final survey. W e select sources from the V IPER S catalogue in the red sh ift range 0 .6 -1 .0 w ith red sh ift confidence > 95% (redshift flags 2,3,4,9). T he redshift distribution is show n in Fig. 1. T he sources have estim ated rest fram e B user B -band m agnitudes and U - V colours defined in the Johnson-C ousins-K ron system as described b y F ritz et al.
(2014) . T he total n um ber o f sources used in th e analysis is 36928. W e construct subsam ples o f galaxies in bins o f redshift (Az = 0.1), lum inosity (A M B = 0.5), and colour as illustrated in Fig. 2 . W e separate red and blue galaxy classes using th e cut defined b y F ritz e t al. (2014) at (M U - M V)Vega + 0.25z = 1.1 in the Vega system . In total there are 37 bins, 19 fo r blu e an d 18 for red galaxy subsam ples, including those that are not com plete, see th e discussion in Sect. 3.3.
2 .2 . S u r v e y c o m p l e t e n e s s
T he V IPER S survey coverage is ch aracterised by an angular m ask (G uzzo e t al. 2014) . T he m ask is m ad e up o f a m o saic o f V IM O S pointings, each consisting o f four quadrants.
R egions around bright stars and o f p o o r photom etric quality in the C F H T LS p hotom etric catalogue have been rem oved.
Fig. 2. Subsamples o f galaxies in colour, absolute magnitude and red
shift bins. Left: the absolute magnitude-colour plane. The histogram at top shows the distribution of colour. The sample is divided into blue and red classes following Fritz et al. (2014) at M j - M V + 0.25z = 1.1. Right:
the sample is further binned by redshift and absolute magnitude. The lu
m inosity bins account for the mean evolutionary trend. The faintest lu
m inosity bins are not volume limited and the thick blue and dashed red curves show the limiting magnitudes for blue and red galaxy classes.
W ithin an observed quadrant there are m any factors includ
ing the intrinsic source properties, instrum ent response an d ob serving conditions that determ ine the final selection function (G arilli e t al. 2014) . T he fraction o f sources out o f the p ar
en t p hotom etric sam ple that are targeted for spectroscopy is re ferred to as the target sam pling rate (TSR ). A m ong the targeted sources, not all w ill give a reliable red sh ift m easurem ent. W e re fer to this fraction as th e spectroscopic success rate (SSR). The sam pling rate is the pro d u ct o f the T S R and SSR: r = r TsRrssR.
T he arrangem ent o f slits in V IM O S is strongly constrained since the spectra cannot overlap on the im aging plane (B ottini e t al. 2005) . In V IPE R S, the resu lt is th at the n um ber o f tar
g eted sources in a pointing is approxim ately constant, dam ping th e g alaxy clustering signal on b oth sm all an d large scales (Pollo e t al. 2 0 0 5 ; de la Torre et al. 2013) .
As d escribed in Sect. 3, in our d ata m odel w e bin th e galaxies o nto a cubic grid, so it is only necessary to estim ate th e T S R cor
rection on th e scale o f the cubic cell. F or 5 h -1 M pc cells, this corresponds to 10 arcm in at z = 0.7, w hich is larger than the V IM O S quadrants. T he T SR is estim ated on a fine grid as the fraction o f targets out o f the p arent sam ple w ithin a 3 arcm in cir
cu lar aperture: Ntai-get/Nparent. T he fine grid is then dow n-sam pled to determ ine the average T SR in each g rid cell. T he colours in F ig. 3 indicate the T SR m easurem ents as a function o f angular position (at the positions o f observed galaxies).
T he spectroscopic success rate is prim arily co rrelated to the conditions a t th e tim e o f observation and so varies w ith pointing.
Fig. 3. Distribution of VIPERS targets on the sky is plotted for the two fields W1 and W4. The points are coloured according to the target sampling rate (TSR) which is defined as the ratio o f the num ber of targeted galaxies in a patch o f sky over the total num ber available in the parent photom etric catalogue. In this w ork we estim ate the TSR w ithin a circular aperture with radius 3 arcmin. The TSR depends on the projected density of targets on the sky. TSR is higher in low-density fields w ith few potential targets while in high-density fields slit positioning constraints severely lim it the num ber of sources that m ay be targeted. The inset histogram shows the cumulative fraction o f targets with TSR below a given value. The median TSR over the survey is 48%.
Fig. 4. The spectroscopic success rate (SSR) quantifies the fraction of targets for w hich the redshift could be m easured with >95% confidence. On the left we show the mean SSR of each pointing divided by quadrant (7 x 8 arcmin). The inset histogram gives the cumulative number o f quadrants with SSR below the given value. On the right, we show the SSR as a function o f iAB m agnitude (solid curves) w ith an analytic fit (dashed curves).
The sample is divided based on the overall quality of the quadrants quantified by SSR. The quadrants are ranked by m ean SSR and the curves are com puted for each decile. The range o f SSR and the number o f sources in the bin are given in the figure legend.
F or a p articular source the SSR depends on the apparent flux as w ell as the spectral features that are available to m a k e the red- shift m easurem ent. W e find that the prim ary contribution com es from the apparent flux, and w e quantify the m ean SSR, defined as N measured/N target in each quadrant, as a function o f th e i b and m agnitude (G uzzo e t al. 2 0 1 4 ; G arilli et al. 2014) . T he deg ra
dation is m o st severe in p o o r observing conditions, so w e co m pute the SSR separately according to the quality o f the quadrant.
W e ran k the quadrants b ased on m ean SSR and com pute sep
arately r SSR(mi) in each decile. T he SSR is fit w ith an analytic form : r SSR(m) = a(1 - e c(m-b)). In the rig h t panel o f Fig. 4 w e see th at the SSR depends strongly on the q uadrant quality. For the top 10% o f quadrants (show n by th e re d curve in Fig. 4 ) , the SSR rem ains > 9 0 % , b u t it drops quickly as quadrant q u al
ity falls. W h at is im portant to no te here is th at the shape o f the r SSR(iAB) curves changes as a function o f q uadrant quality.
T he effect o f the w eights on th e red sh ift distribution is shown in F ig. 1.
2 .3 . M o c k c a ta l o g u e s
W e use a set o f sim ulated (m ock) galaxy catalogues constructed to m a tch the V IPER S observing strategy. T he catalogues are b u ilt on the M ultiD ark N -b o d y sim ulation (P rada e t al. 2012) u s
ing the H alo O ccupation D istribution (H O D ) technique. D etails o f th e construction m ay b e found in (d e la Torre et al. 2 0 1 3 ; de la Torre & P eacock 2013) .
G alaxies w ere added to the d ark m a tte r halos in th e sim ula
tion according to a lum inosity dependent H O D m odel. T he cor
rela tio n function and n um ber counts in lum inosity bins w ere set to m atch m easurem ents m ade at 0.5 < z < 1.2 in C FH T LS, V V D S and earlier releases o f V IPE R S. E ach m o c k galaxy is
characterised b y its an g u lar coordinate, com oving distance, o b served red sh ift including its errors and an absolute m agnitude in the B band.
W e p artition th e m o ck catalogues into bins o f red sh ift and lum inosity, b u t n o t in colour, as w e do for the V PER S data. The step size in red sh ift and lum inosity are Az = 0.1 and AM B = 0.2 over the red sh ift ran g e 0.6 < z < 1.0. W e have 46 bins including those bins th at are n o t com plete ow ing to the apparent flux lim it.
T he m o ck galaxies m atch the nu m b er density o f the V IPER S observations, b u t do n o t include the slit placem en t constraints th at w e correct in the d ata w ith the T SR w eights. T he catalogues do n o t sim ulate th e spectroscopic sam pling rate.
3. Data model
3.1. G a la xy n u m b e r c o u n ts
W e overlay a three-dim ensional cartesian g rid on the survey. The n um ber o f galaxies in a given sam ple observed w ithin a cell in dexed b y i is related to an underlying continuous galaxy density field 6 G by
N i = N Wi(1 + 6G,i) + £i, (2)
w here wi is the spatial selection function and N i is the m ean d en sity providing the norm alisation. T he stochastic n atu re o f galaxy counts is captured b y the ran d o m variable e* an d is dom inated by Poisson noise except in the highest density peaks (D i P orto et al. 2014) . T he cells are defined in com oving redshift-space coordinates and w e ado p t a fiducial cosm ology to define th e re lationship betw een red sh ift and com oving distance.
T he selection function w in Eq. (2) gives the likelihood o f observing a galaxy at a given grid point. It accounts fo r the an gular geom etry o f the survey, sam pling rate and red sh ift d istri
bution. In this analysis w e separate th e an g u lar and line-of-sight com ponents. As described in Sect. 5.1, the angular dependence is determ ined from the survey m ask and T SR w hile the redshift distribution is com puted assum ing the lum inosity function and apparent flux lim it fo r the given subsam ple o f galaxies.
T he expected n um ber o f galaxies in a cell is given by the pro d u ct o f N and the selection function, giving
(Ni) = N wi. (3)
T he selection function described can account for spatial v aria
tions but cannot describe sam pling dependencies on galaxy type o r apparent flux. F o r V IPER S data w e w ill u p-w eight galaxies b ased on the inverse SSR depending on quadrant an d apparent i b an d m agnitude. T hese w eights are only indirectly correlated w ith the density field so they resu lt in an am plification o f the shot n oise level. T he SSR w eig h t o f a galaxy is wSSR and the noise am plification factor is a = (wSSR) averaged over all galaxies in the subsam ple. T herefore, in Eq. (2 ) N i represents the w eighted co u n t o f galaxies and, consequently, the variance o f the stochas
tic term , ^ , is b o o sted to arn2e.
In this analysis w e discretise the galaxy field onto a coarse spatial grid as w ell as onto a finite grid o f F o u rier m odes. This process introduces an error in the density field arising from the aliasing o f structures: sm all-scale structures w ith spatial fre
quencies h igher than the N y q u ist frequency becom e im printed on larger scales (H ockney & E astw ood 1988) . T he effect m ay b e corrected for in th e pow er spectrum b y assum ing the spec
tral shape above the N y q u ist frequency (Jing 2005). However, to accurately reco n stru ct the density field w ithout m aking such assum ptions, w e m ay use a m ass-assignm ent schem e o r an ti
aliasing filter discussed in A ppendix A . T he sm oothing effect
o f m ass-assignm ent schem es introduces a convolution in E q. (2) w hich invalidates the sim ple count m odel. A n alternative is to use th e super-sam pling m ethod of Jasche e t al. (2009) th at ap proxim ates the ideal anti-aliasing filter and does n o t dam p sm all- scale power. Since w e desire a com pact w indow in b oth configu
ratio n and F ourier space, w e ad ap t this technique w ith a soft cu t as described in A ppendix A . This approach reduces the aliased signal to the level o f the triangle-shaped-cell schem e w hile p re serving sm all-scale pow er to k ~ 0.7kNyquist.
T he convolution introduced by the anti-aliasing filter m o d i
fies the n o ise properties such th at th e Poisson expectation, <r2 = N iw , cannot b e assum ed. Instead w e use a re-scaled Poisson variance ch aracterised by the factor v* = o f / N As described in Sect. 5, th e factor m ay be estim ated in a M onte C arlo fashion given th e m ask and anti-aliasing filter.
A cell p artially cu t by the m ask w ill becom e strongly coupled w ith its n eig h b o u r through the anti-aliasing filter. In practice w e neg lect th e additional off-diagonal cell-cell contributions in the n o ise covariance m atrix. H owever, w e found that it is necessary to regularise th e n oise m atrix b y increasing the n oise level to achieve stable results. W e set a low er lim it on the cell variance through a param eter threshold such th at the scaled shot n o ise has a floor set by v* ^ m ax(vthreshoid, v*). T he p rocedure is identically applied b oth to m o ck and real data.
3.2. G a la xy b ia s
W e assum e a constant linear biasing m o d el such that
6 G,i = bD (zi )6i, (4)
w here z* is the red sh ift o f the cell indexed by i. T he bias factor b depends on the lum inosity and colour o f th e galaxy subsam ple.
W e give explicitly the grow th o f m a tte r fluctuations w ith tim e according to the linear grow th factor D(z) = D (z )/D ( z ref) w ith 6(z) = 6(zref)D (z). F o r the V IPER S sam ple the reference red sh ift is set to zref = 0.7. Since V IPER S covers an extended red sh ift ran g e this facto r brings large-scale density m odes to a com m on epoch.
3.3. N u m b e r d e n s ity
T he n u m b er density o f galaxies in a lum inosity b in is given by th e integral o f the lum inosity function:
n Mfaint(z)
n(z) = n(M , z)dM . (5)
^Mbright
W e param eterise th e lum inosity function using the S chechter function (S chechter 1976) in term s o f m agnitudes and the p a ram eters (0*, M *, a):
n(M ) = 0.4 ln 10 0* (1 0 0-4(M*-M))a+1 exp (10°'4(M*-M)) . (6) T he characteristic m agnitude evolves as M *(z) = M *(0) + Ez w ith E x - 1 for red galaxies in V IPER S (F ritz e t al. 2014) confirm ing the findings o f previous studies at m o derate red sh ift (Ilb ert et al. 2 0 0 5 ; Z u cc a e t al. 2009) .
T he nu m b er density observed is further reduced by th e sur
vey com pleteness. In V IPE R S, galaxies are targeted to an appar
en t m agnitude lim it o f mlim = 22.5 in th e iAB photom etric band.
This sets an absolute m agnitude lim it for a given class o f galaxy
Mlimit(z) = mlim - Dm(z) - K (z ), (7)
w here Dm(z) is th e distance m odulus w hich depends only on the b ackground cosm ology an d K is the K -correction term w hich depends on th e particu lar type o f galaxy targeted. T he absolute m agnitudes w ere com puted for each galaxy by fitting spectral energy distribution tem plates to b roadband photom etry as d e
scribed b y D avidzon e t al. (2013) and from the absolute m a g n i
tudes w e infer the K -corrections. W e p aram eterise the tren d o f K -correction w ith red sh ift as K (z ) = K +AK(z). T he value o f K 0 is estim ated from the m edian value o f galaxies w ithin a given subsam ple w hile AK(z) is a p olynom ial fit w ith the follow ing coefficients, different fo r red and blu e galaxy types:
A ^b lu efe) = 1.784(z - 0.7)2 + 0.440(z - 0.7) - 0.678 (8) AK ,re d(z) = 2.144(z - 0.7)2 + 1.745(z - 0.7) - 0.720. (9) U sing this param eterisation the inferred m agnitude lim its cor
responding to 50% com pleteness are indicated in Fig. 2 for the blue and re d sam ples by the solid an d d ashed lines. For m ock sam ples, the K -correction term and its evolution are fixed to Km ock = z - 1.3 .
W ith these ingredients w e m o d el the red sh ift distribution o f each galaxy subsam ple by integrating the lum inosity function w ith Eqs. (7) an d (5) . W e leave the m ean density o f each subsam p le free to set th e n orm alisation o f th e red sh ift distribution. W e then take the shape given by the S chechter function to in terp o late th e lum inosity function across the bin. T he param eters M*
and a in each colour and red sh ift bin are fixed to the values m e a
sured in V IPER S (F ritz et al. 2014) . S ince the p recise lum inosity evolution is n o t know n, the evolution term , £ , is allow ed to vary as a function o f colour an d redshift. This gives a characteristic m agnitude M*(z) = M*(zr e f) + £ ( z - zre f), w here zref is taken to be the m idpoint o f the red sh ift bin. C hanging £ m odifies th e shape o f the red sh ift distribution.
3 .4 . P o w e r s p e c tr u m
T he m atter pow er spectrum in real space P (k) = <|6k|2} is a s
sum ed to b e isotropic. Seen in redshift-space, it is distorted along the line-of-sight direction (H am ilton 1998) . W e m odel the signal on the cartesian F ourier grid as
Table 1. Accounting of the free param eters in the data model.
Param eter Symbol D imension
Overdensity field 6 2 x 72 x 16 x 172 (5 h -1 Mpc cubic cells)
Power spectrum P 109
Distortion factor 1
Velocity dispersion 1
Galaxy bias b 37 (19 blue, 18 red)
Mean num ber density IV 37
Lum inosity evolution £ 8
w here u = ki o s/k an d k = ^ k 2x + k2 + k2z . T he line-of-sight d i
rection is aligned w ith the grid such that klo s = kz taking the p lane-parallel approxim ation.
T he coherent m otions o f galaxies on large scales are d e
scribed by the K aiser ( 1987) factor w ith f i = f / bg, w here the grow th rate in A C D M is f ( z ) = d lo g D / d log a. O n sm all scales, velocities random ise and m ay be m odelled b y an ex p onential p airw ise velocity dispersion giving a L orentzian p ro file in F ourier space w hich w e refer to as the dispersion m odel (B allinger et al. 1996) . T he velocity dispersion term , in Eq. ( 10) , has units o f h -1 M pc. T he conversion to velocity units is H (z )/(1 + z ) /V 2 ~ 60.0 h M p c-1 k m s -1, w hich is nearly constant over the red sh ift ran g e o f interest. W e add a G aussian term along th e line o f sight to characterise redshift m easurem ent errors w here ^ob s = &cz/H (z ) an d <rcz is the red sh ift error. For V IPER S the estim ated red sh ift error is <rcz = 141(1 + z) k m s -1
(G uzzo et al. 2014) and ^o b s = 1.67 h -1M p c an d is nearly constant over the red sh ift ran g e 0 .6 -1 .0 .
T he factor B (k x, ky, kz) accounts for th e cell w indow function arising from the anti-aliasing filter and is given by Eq. (A .2) . In this analysis the absolute am plitude o f the pow er spectrum is n o t constrained. So w e set the am plitude A in Eq. ( 10) to fix ^ 8 = 0.8, th e variance com puted on a scale o f R = 8 h-1 M p c integrated to th e N yquist frequency.
W e ignore geom etric distortions arising from the choice o f th e fiducial cosm ology (A lcock & P aczynski 1979) . T he resu lt
ing bias is n o t significant w hen com pared w ith the statistical uncertainties o f th e V IPER S redshift-space clustering m easu re
m ents (de la Torre e t al. 2013) . H owever, w hen carrying out a m o d el test, w e m ay rescale the density field an d tw o p oin t statis
tics to transform from the fiducial to the test cosm ology as car
ried o u t for the V IPER S pow er spectrum analysis by R ota et al.
(in prep.), but this is n o t done here.
T hese steps are rep eated form ing a M arkov chain and after an in itial burn-in p eriod w e can expect that the sam ples are rep re
sentative o f the jo in t p osterior distribution.
In th e first step, w e sam ple from th e conditional pro b ab il
ity distribution for the density field in a tw o-stage procedure.
F irst, the W iener filter is used to com pute the m axim um a priori field bWF (K itaura e t al. 2010) . T he W iener filter solution is a sm oothed field th at gives an u nderestim ate o f th e true pow er. To g enerate a realisation o f the density field a ran d o m com ponent th at is uncorrelated w ith the observations brandom is added (Jew ell e t al. 2004) . T he final field is thus the sum 6 = bWF + brandom.
A fter constructing a realisation o f the density field, the sec
on d step is to sam ple th e pow er spectrum . W e p u t a G aussian p rio r on the first bin at k < 0.01 h M p c -1 setting the m ean and variance to the fiducial value and sam ple variance expectation.
( i + j3/j,2)2 k2
S ( k , K p , (TV, ^obs) = a 1 \ e - “ B 2(kx, ky, kz)P (k), (10) + lo s ^
4. Gibbs sampler
W e presen t a b rie f overview o f the G ibbs sam pler. S ince ou r im plem entation differs from th at o f Jasche & W andelt (2013b) w e provide a detailed d escription in A ppendix B . T he full param eter set introduced in the previous section is sum m arised in Table 1 . W e use the G ibbs sam pling m eth o d to sam ple from th e jo in t p o s
terio r o f the p aram eter set. This is p erform ed b y iteratively draw ing sam ples from each conditional p robability distribution in the follow ing steps (w here v in d ic a te s that a sam ple is draw n from th e given distribution):
1. generate ds+i v p(<5|iVs, bs, P s, p s,<rsv, N );
2. generate P s+1 v p ( P |d s+ \ N s, b s, p s,( r sv, N );
3. norm alise pow er spectrum P s+1;
4. g e n e ra te p s+l ,o-sv+l v p ( p , a vIPs+l , 5 s+l , N s, b s, N );
5. generate N ^ 1 v p(JV|bs+ \ P s+i ,J6 s+i ,^V+i , ^ s+i , N );
6. generate £ s+1 v p(£|JV s, bs+1, P s+i ,J6s+i , ^ vs+i , d s+i , N );
7. generate b s+1 v p (b lP s+1 , p s+1 ,^ ^ + 1, 6 s+1, N s+ \ N ) .
This aids th e stability o f the chain. A uniform p rio r is used for the bins at k > 0.01 h M p c-1 . W e use tw o approaches to sam ple the po w er spectrum d etailed in A ppendix B. . F irst, w e draw sam ples from the inverse-gam m a distribution, see e.g. Jasche et al. (2010b) ; however, this produces very sm all steps in the low signal-to-noise regim e and so can be inefficient at sm all scales. T herefore, on alternative steps w e carry out a M etropolis- H astings routine to draw sam ples o f the p ow er spectrum acco rd ing to the likelihood, Eq. (B .2). W e find consistent sam pling o f the pow er spectrum using the tw o m ethods.
S ince w e cannot constrain th e absolute norm alisation o f the pow er spectrum , w e norm alise to the desired value o f ^ 8. W e then draw the redshift-space distortion param eters j3, w hich are independent o f the p ow er spectrum am plitude.
N ext, w e sam ple from the m ean density conditional p ro b ability distribution fo r each galaxy sam ple w hich includes the evolution factor E. H ere w e use a Poisson distribution, as described in A ppendix B .4 .
F inally w e sam ple from the bias conditional p robability d is
tribution fo r each galaxy sam ple. This distribution is G aussian fo r the bias param eter (see A ppendix B .3) . In this m ethod, the bias is com puted on the redshift-space grid, w hich in ou r case has a resolution o f 5 h -1 M pc. F or physical interpretation it is interesting to estim ate th e bias averaged on larger scales. So, in estim ating th e bias w e first dow n-grade the grid resolution by a factor o f tw o, such th at the bias is averaged over a scale o f 10 h -1 M pc. W e im pose a uniform p rio r for th e bias values o f 0.5 < b < 4.
5. Application to VIPERS
5.1. S e t-u p
T he data and m o ck catalogues are p rocessed sim ilarly, although the construction o f galaxy subsam ples differs. T he m o c k ca ta
logues do n o t include the inhom ogeneous incom pleteness cor
rected fo r in th e data by the S SR and T SR factors. T he un cer
tainties introduced b y these corrections are negligible com pared w ith statistical uncertainties in V IPE R S.
1. T he tw o survey fields, W 1 an d W 4 are separately em bedded into rectangular boxes. T he grids have dim ensions 72 x 16 x 172 cells and each cubic cell has com oving size 5 h -1 M pc.
W e align th e grid such th at a t the field centre the three axes correspond to the rig h t ascension, declination and line-of- sight directions. T he co-m oving coordinates are com puted using the fiducial cosm ological m odel. In the real catalogue alone, galaxies are up-w eighted b y the inverse SSR dep en d ing on quadrant and apparent i-band m agnitude.
2. W e com pute the density on the grid using the anti-aliasing filter b ased on the super-sam pling m eth o d p roposed by Jasche e t al. (2009) w ith a soft k-space cut-off as described in A ppendix A .
3. T he angular ( a , 6) and radial (z) com ponents o f th e selection function are com puted separately on the grid: w (ai, 6i, zi) = w (ai, 6i)w(zi).
4. F o r the angular com ponent, w e generate a uniform grid o f test points th at over-sam ple the grid by a facto r o f 8 and reje ct points outside the survey an g u lar m ask. F o r V IPER S d ata the rem aining are dow n-sam pled by the TSR . T he points are then assigned to the g rid points using th e anti-aliasing m ass-assignm ent schem e. T he selection function w* is then given b y the norm alised density o f test p oints on the grid.
5. T he radial com ponent o f the selection function is estim ated in bins o f redshift, lum inosity, and colour. W e estim ate the
m edian K -correction term for galaxies w ithin each bin and use E q. (5) to com pute the unnorm alised N (z).
6. W e estim ate th e generalised shot n o ise variance v* = / TVj w hich depends on th e m ask through the anti-aliasing filter in M onte C arlo fashion. W e generate a set o f 1000 shot noise m aps b y distributing ran d o m points over the survey volum e.
F o r V IPER S data the points are dow n-sam pled b y TSR . W e then com pute the variance for each cell o f the m ap over the 1000 realisations. To regularise th e noise covariance m atrix, a threshold is set v* = max(vthresh, v*), w here vthresh = 0.3 for m ocks an d 0.15 for data to account for TSR.
W ith the galaxy n u m b er density m ap, selection function and n o ise m ap, w e have all the com ponents o f th e data m odel req u ired to estim ate the p o sterio r p robability distribution in E q. ( 1) . To sam ple this probability distribution w e ru n th e G ibbs sam pler M arkov chain for 2000 steps and, allow ing fo r a burn- in period, begin th e analysis from step 1000. T he convergence properties and ju stification fo r th e burn-in perio d are show n in A ppendix C . F o r the V IPER S data w e ran seven independent chains for 2000 steps each providing 7000 post-burn-in sam ples fo r analysis.
Taking the variance o f the M arkov chain gives us an internal error estim ate on the param eters. T he runs on m o c k catalogues show th at the chain variance corresponds to the expected sam ple variance for the pow er spectrum . H owever, this is n o t necessarily true for the o ther statistics. F or instance the lum inosity function quantifies the distribution o f observed galaxies th at rem ains fixed in the chain. It is only indirectly d ependent on the underlying density.
5.2. D e n sity field
T he density field taken from a single step (1500) in the M arkov chain is show n in Fig. 5 . It represents the application o f the W iener filter on a b ias-w eighted com bination o f the galaxy sub
sam ples. T he reconstruction is b ased on the redshift-space d is
tortion m odel and so the resulting field is anisotropic an d is ch aracterised by effective redshift-space distortion param eters averaged o ver the galaxy sam ples.
T he resu lt o f the W iener filter is an adaptively sm oothed field th at extrapolates structures o ver the correlation length o f a few m egaparsecs. To build a full realisation o f the structures w e ad d a G aussian constrained realisation th at fills in the gaps and gives the full variance. T he structures outside the survey bo u n d ary are g enerated from a ran d o m G aussian realisation although th e p hases are p roperly aligned a t the boundary. T he true galaxy density field on scales o f 5 h M p c-1 is far from G aussian and the difference is v isible b y eye.
In F ig. 5 w e can recognise the cosm ic w eb o f structures including knots, filam ents an d void regions. T he structures are rich e st w here the sam pling is highest a t low er redshift. A t red- shift z > 0.8 w e see few er coherent structures and the co n tri
bution from the constrained G aussian realisation is larger. E ach step o f the M arkov chain gives a reconstruction o f the field w ith different realisations o f n oise and large-scale m odes. O nce the chain has passed the burn-in p eriod (see A ppendix C ), w e can co n sid er these realisations to rep resen t G aussian perturbations around the observed galaxy field.
5.3. R e d s h ift-s p a c e p o w e r s p e c tr u m
T he galaxy po w er spectrum in red sh ift space is param eterised in term s o f the real-space m a tte r po w er spectrum , bias, and
F ig.5. VIPERS cone diagrams for the fields W1 (top) and W 4 (bottom). The left panels show the redshift-space positions o f observed galaxies.
The marker colour indicates the blue or red colour class and the marker size scales w ith B-band luminosity. The depth o f the slice is 10 h -1 Mpc.
The orange line traces the field boundaries cut in the redshift direction at 0.6 < z < 1.0. A t right we show a slice o f the density field taken from one step in the Markov chain. It represents the anisotropic W iener reconstruction from the w eighted com bination of galaxy tracers. The field is filled with a constrained Gaussian realisation. The field has been smoothed w ith a Gaussian kernel with a full w idth half m axim um of 10 h -1 Mpc.
The colour scale gives the over-density value.
Fig. 6. Constraints on the real-space pow er spectrum. The low er panel shows the relative difference with the fiducial model. The black dots give our estim ates of the binned real-space power spectrum taken from the median o f the Markov chain. Overplotted is the fiducial m odel adopted in this study (black dashed curve). We find agreem ent w ith the best-fit m odel using VIPERS data by Rota et al. (in prep.) (purple dot-dashed curve). The pink dashed curve is the mean of the power spectrum estimates taken from the 27 m ock catalogues. We present three error estimates:
the internal chain variance determined from VIPERS data (grey steps), the chain variance determined from m ock catalogues (blue steps) and the variance o f the individual estim ates from the 27 m ock catalogues (red steps). The error corridors show 70% confidence intervals.
redshift-space distortion factors (Eq. ( 10)). W e bin the pow er spectrum linearly w ith b in size Ak = 0.01 giving 109 bins. The redshift-space distortion param eters are fit to k < 0.4 h M p c-1 . This lim it corresponds to k a v x 1 w here w e can expect the dispersion m odel to b reak dow n.
T he M arkov chain provides jo in t sam ples o f th e p ara m e
ters. In F ig. 6 w e show the m edian over the pow er spectrum chain (black dots). T he confidence cooridor gives the 1 ^ co n fidence interval estim ated from th e chain variance (grey steps).
W e find good agreem ent w ith the m odel com puted w ith CLA SS an d H alofit (black dashed) w ith Q m = 0.27 (L esgourgues 2 0 1 1 ; S m ith e t al. 2 0 0 3 ; Takahashi e t al. 2012) . T he best-fitting m odel determ ined by R ota et al. (in prep.) has Q m = 0.272 ± .03 (over-plotted w ith purple d ot-dashed curve).
O n sm all scales k > 0.45 h M p c -1 (0 .7 5 x N yquist frequency) th e pow er drops. This is due to neglecting correlations betw een cells th at arise b ecause o f the anti-aliasing filter. O n large scales, there is a dip in pow er at k = 0.05 h M p c-1 seen in b o th m ock
Table 2. Constraints on redshift-space distortion parameters.
P
chain % 6,m ock b eff ^b,chain ^b,mock f ^8 ^ f , chain ^ f,m ockM ock 0.47 -0 .1 2 /+ 0 .0 9 0.09 1.54 -0 .0 4 /+ 0 .0 4 0.03 0.46 + 0 .0 9 /-0 .1 2 0.08 VIPERS 0.41 -0 .0 8 /+ 0 .0 7 1 .44 -0 .0 3 /+ 0 .0 2 0.38 -0 .0 7 /+ 0 .0 6
Notes. We give the 68% confidence intervals from the chains and the standard deviation among the 26 m ock catalogues. The fiducial value is f ^ ( z = 0.7) = 0.45.
catalogues and data, although it biases the estim ate only at the level. Scales at k < 0.06 h M p c-1 are only m easu red in the line-of-sight direction w ith V IPER S so th e inability to rec o n struct them properly w ithout a p rio r constraint is n o t surprising.
T he m edian values w e find for the redshift-space distortion param eters are jdV IPE R S = 0.41 and j6m ock = 0.47. T he w ithin- chain variance is ^ch ain = (-0 .1 2 , +0.10) (68% confidence inter
val), w hile th e scatter o f th e 2 6 m ocks gives standard deviation
V m ock = ° .° 9 .
W e com pute th e grow th ra te through th e relatio n
f c T 8 (z ) = ^ 8 , g a l a x y (zX
-+0.09 -0.12
(11) w here ^ 8,gaiaxy = beff^ 8. In this analysis w e have fixed the a m p litude o f the m a tte r pow er spectrum a t red sh ift z = 0.7 w ith a 8(z = 0.7) = 0.643 w hich corresponds to ^ 8 = 0.8 a t z = 0 in the fiducial cosm ology.
W e com pute the effective galaxy bias as the num ber-w eighted average over the galaxy sam ples,
(
12)
w here the sum s are over the galaxy subsam ples an d selection function grids.
W e sum m arise the constraints on the grow th ra te in Table 2 . W e find (f^8)vIPERS = 0.38+(1-(17 and ( f ^ W k = 0.46 at z = 0.7 w here w e quote th e chain variance. T he scatter betw een the m ocks gives a standard deviation <rf,m ock = 0.08. Thus these constraints on the grow th rate are in agreem ent w ith the V IPER S correlation function m easurem ent b y de la Torre et al. (2 013) . T here, the error w as 16% on the grow th rate f< r8 = 0.48 a t z = 0.8. In this w ork w e find an error o f 18%. W e attribute the higher error in this analysis to the fact th at w e m arginalise over the real- space pow er spectrum , w hile in the previous analysis it was fixed to a fiducial cosm ology.
T he correlations betw een a subset o f the pow er spectrum and redshift-space d istortion param eters are show n in Fig. 7 . T he star sym bols m a rk the m e d ian values o f th e param eters estim ated from the V IPER S M arkov chain w hile the filled contours give the 70% and 90% confidence intervals. T he m edian value and m arg in alised 70% uncertainty on each p aram eter are labelled.
W e find that f< r8 (18% relative error) is b etter constrained than P (20% error). This is due to the anti-correlation betw een beff and P w ith correlation co efficien tp = - 0 .5 8 . T hese correlations arise from th e specific p aram eterisation adopted and w ould be m o d i
fied under a different d ata m odel.
T he b lack dots in F ig. 7 represent the m edian values esti
m a te d from individual m o c k catalogues. W e find that the value o f th e galaxy bias is different w ithin the m ocks (beff = 1.55) and V IPER S (beff = 1.44). T he bias o f th e m o c k galaxies is determ ined by th e lu m inosity-dependent H O D prescription (de la Torre e t al. 2013) an d so the m inor difference from real d ata is n o t u nexpected. A ccounting for the difference in bias, w e find excellent agreem ent betw een the distribution o f m ocks and
Fig. 7. Degeneracies between RSD param eters yS, (xv, e ffective bias and the power spectrum at k = 0.4 h M pc-1. The shaded regions m ark the 68% and 95% confidence intervals from VIPERS chain and the star symbols m ark the m ean value. The points give the distribution of mean values derived from m ock catalogues. The histogram s along the diag
onal give the marginalised distributions of each param eter chain. The filled histogram gives the distribution from the VIPERS chain, while the solid line is the distribution o f mean values derived from the mock catalogues.
th e param eters estim ated from real data. T he sim ilarity o f the p robability distribution function shapes also gives us confidence in the analysis m eth o d and error estim ates. F urtherm ore, since th e m o ck s do n o t include m a n y o f the selection effects presen t in th e d ata the agreem ent suggests th at these sources o f system atic uncertainties do n o t influence our conclusions.
5.4. C olour a n d lu m in o sity d e p e n d e n t g a la x y b ia s
W e com pute the galaxy bias from the variance o f the galaxy counts on the grid. H ow ever, w e first dow n-sam ple th e grid b y a factor o f tw o such th at th e bias is com puted on a scale o f 10 h -1 M pc.
In Fig. 8 w e show the m e d ian bias values o f th e M arkov chain and the confidence intervals are given b y th e chain vari
ance. T he bias is com puted in bins o f redshift, lum inosity, and colour. W e find a colour bim odality w ith re d galaxies m o re strongly b ia sed than blue. This corresponds to th e w ell-know n g alaxy m o rp hology-density relatio n sh ip th at early type galax
ies are predom inantly found in high density environm ents (e.g.
C ucciati et al. 2 0 0 6 ; D ressler 1980; D avis & G eller 1976) . ,2 _ H l,jNlW i,itf
eff _ Z i,iŃ w i,i ,
indicates th at the V IPER S volum e is large enough such th at the sam ple variance does n o t significantly alter the am plitude and th at any effects due to the lum inosity dependence o f bias are w eak.
5.6. P a ra m e te r co va ria n ce
W e estim ate the covariance o f the statistics w ith the M arkov chain. F igure 1 1 show s the norm alised correlation m atrix deter
m in ed in ou r analysis, 2 Cij
' » = ^
(13)
Fig. 8. VIPERS galaxy bias parameters in redshift, luminosity, and colour bins. A colour bimodality is seen in each redshift bin. The trend with luminosity is most striking in the lower redshift bins for both blue and red galaxies.
Sim ilarly, w e expect to find that galaxy bias increases w ith galaxy lum inosity since m o re m assive and m o re lum inous g alax ies tend to form in m o re m assive dark m atter clum ps (C oupon e t a l. 2012) .
P revious studies w ith V IPER S data estim ated the g alaxy bias o f the full galaxy sam ple as a function o f lum inosity and redshift.
M arulli e t al. (2013) m easu red the pro jected galaxy correlation function in lum inosity an d red sh ift bins to constrain th e m ean bias averaged over scales 5 -2 0 h -1 M pc. D i Porto et al. (2014) m odelled the counts-in-cells probability distribution function to estim ate the linear bias. To com pare our estim ate o f the galaxy bias w ith these previous results w e construct lum inosity th resh old sam ples counting both red and blu e galaxies. W e cro ss
correlate these n um ber density m aps w ith the W iener density field from th e core analysis and estim ate th e m easurem ent u n certainty from the chain variance. T he resulting bias values are show n by the m arkers w ith error bars in Fig. 9 . W e find excel
len t agreem ent w ith the p revious analyses, although o ur redshift bins differ. T he bias values from D i P orto e t al. (2014) have been taken on a scale R = 8 h -1 M p c w hile those o f M arulli et al. (2 013) are sensitive to sm aller scales. T he disagreem ent at z > 0.9 m ay indicate th at the bias o f lum inous galaxies is scale dependent a t high redshifts p robed differently by the three studies.
5.5. L u m in o sity function
In F ig. 10 w e show the derived lum inosity function b ased on the m ean galaxy n um ber density (dots w ith error bars) for different galaxy types. W e com pare the resu lt to the analysis from Fritz et al. (
2014
) b ased on the S andage-Tam m ann-Y ahil (STY, Ilbert et al. 2 0 0 5 ; Sandage e t al. 1979) estim ator (dashed curves). W e can expect to find a difference in th e tw o estim ates arising from how the galaxy bias is treated. T he S TY estim ate is designed to b e independent o f th e underlying density field u nder the approxim ation th at th e galaxy lum inosity is uncorrelated w ith density.
A strong lum inosity dependence o f the bias can system atically tilt the inferred lum inosity function (Sm ith 2 0 1 2 ; C ole 2011) . T he agreem ent b etw een ou r analysis and the STY m easurem ent
T he bias and m ean nu m b er density param eters are ordered first b y lum inosity and colour and then b y red sh ift bin. T he appear
ance o f blocks in the m atrix indicates th at w ithin red sh ift bins the statistics are strongly correlated. W e also find th at bias and m ean density are anti-correlated, th at is, increasing bias necessitates decreasing m ean density to preserve the sam e fluctuation. The bias an d m ean density p aram eters are w eakly correlated w ith the pow er spectrum m easurem ent.
T he bins o f the pow er spectrum (spacing A k = 0.01 h M p c-1) show independence on large scales, as is expected for the G aussian d ata m odel, b u t they becom e co rrelated at k >
0.3 h M p c-1 . T he correlations arise from the redshift-space p aram etrisation th at couples the am plitude o f the pow er spec
trum to j3 and <rv. T he upper right square in the figure represents th e nearly 100% correlation betw een these tw o param eters.
6. Conclusions
U sing V IPER S w e have dem onstrated a m eth o d o f reconstruct
ing the galaxy density field jo in tly w ith the red sh ift-sp ace pow er spectrum , galaxy biasing function and galaxy lum inosity func
tion w ith m inim al priors on these param eters. T he B ayesian fram ew ork n aturally accounts for the correlations betw een these observables. W e adopt a likelihood function for the galaxy n u m b e r counts th at is given b y a m ultivariate G aussian and set a G aussian prio r on th e density field. T he solution th at m axim ises th e p osterior distribution is given b y the classical W iener filter.
To sam ple from th e po sterio r distribution w e add a G aussian constrained realisation. Incorporating this density field estim a
to r w ithin a G ibbs sam pler, w e jo in tly sam ple the full posterior distribution including the pow er spectrum , bias and lum in o s
ity function param eters. W e find encouraging results by using a m ultivariate G aussian m odel for the likelihood and p rio r d istri
butions, although m o re theoretically m otivated descriptions m ay b e used (K itaura e t al. 2 0 1 2 a; Jasch e & W andelt 2013a) .
T here are clear gains w hen jo in tly estim ating correlated p a ram eters. F o r instance the galaxy colour-density relation can b e used to im prove estim ates o f the density field. F urtherm ore it is w ell know n th at bias w eighting galaxies w hen estim ating th e pow er spectrum leads to im proved accuracy (Percival et al.
2004) and greater statistical pow er (C ai et al. 2011) .
M oreover, the B ayesian fram ew ork provides a recip e for propagating uncertainties in the m easurem ent, incorporating p rio r know ledge and constraints from the data, an d it g uaran
tees reliable error estim ates. In V IPER S w e account for inho- m ogeneous sam pling and detailed angular m asks. W e correct for th e selection function o f V IPER S by up-w eighting galaxies according to th e m a gnitude-dependent spectroscopic sam pling rate, w hile including the target sam pling ra te in the angular d e
pen d en ce o f the survey selection function. T hese corrections are
Fig. 10. Galaxy luminosity function inferred from the m ean density Markov chain for red, blue and combined samples in redshift bins.
Markers are plotted at the median value o f the chain and the height o f the rectangles indicates the 68% confidence interval. The Schechter function fits from Fritz et al. (2014) are overplotted for comparison.
Fig. 11. N orm alised correlation matrix o f the param eters computed from the VIPERS Markov chain. The blocks represent the m ean den
sity, galaxy bias, pow er spectrum and RSD param eters. The structure in the covariance arises from the data m odel param eterisation. The values o f lum inosity and colour dependencies o f galaxy bias and m ean den
sity w ithin a redshift bin are strongly correlated, while they are only w eakly correlated across redshift. On large scales the pow er spectrum covariance is diagonal, but at k > 0.3 h M pc-1 the bins become corre
lated owing to coupling of the small-scale pow er w ith the redshift-space distortion parameters.
fixed in ou r analysis, although fo r upcom ing surveys it w ill be im portant to p ropagate the uncertainties in the selection function to the data products.
Investigating the covariances betw een param eters, w e find strong correlations betw een galaxy bias an d n u m b er density p a ram eters w ithin a given red sh ift bin. This is n o t unexpected since both these param eters dep en d on the one-point p robability d is
tribution function o f the density field. O n the o th e r h and the correlation w ith the p o w er spectrum is w eak.
O u r estim ate o f the po w er spectrum is effectively d econ
volved from the survey w indow function (see R o ta e t al., in prep.) an d w e find th at the covariance o f the pow er spectrum bins is diagonal on large scales as expected from an unm asked G aussian random field. O n sm all scales, k > 0.3 h M p c-1 w e find significant correlations betw een po w er spectrum bins. O n these scales correlations are expected due to the p hysical processes o f structure form ation; however, in this case the correlations
arise from th e param eterisation o f the data m odel. T here is a degeneracy betw een the redshift-space distortion factors j3 and an d the am plitude o f the pow er spectrum on sm all scales.
N evertheless, the erro r estim ate given b y the G ibbs sam pler closely m atches the expectation o f cosm ic variance estim ated from m ock catalogues.
O ur results are in good agreem ent w ith previous V IPER S m easurem ents. W e find values o f th e redshift-space distortion factor j3 th at are consistent w ith the correlation function an aly sis (de la Torre e t al. 2013) . O ur values o f lum inosity dependent bias follow the trends expected from M arulli et al. (2013) and D i Porto e t al. (2014) a t z < 0.9. W e further estim ate the galaxy bias fo r colour sam ples finding a m ore p ronounced dependence on lum inosity for red galaxies than blue. T he lum inosity func
tion w e infer from the m ean n u m b er density m atches w ell w ith those fo u n d by F ritz e t al. (2014) using the STY estim ator.
Fig. 9. Galaxy bias measured from the full (red and blue combined) VIPERS galaxy sample in luminosity threshold bins. Reference data are taken from the VIPERS projected correlation function analysis (Marulli et al. 2013) and counts-in-cells probability distribution function analysis (Di Porto et al. 2014). We note that the redshift ranges differ.
